Question

If ${\log }_{x}x-{\log }_{x}y=a,{\log }_{x}y-{\log }_{x}z=b\&{\log }_{x}z-{\log }_{x}x=C$ , then find the value ${\left(\frac{x}{y}\right)}^{b-c}\times {\left(\frac{y}{z}\right)}^{c-a}\times {\left(\frac{z}{x}\right)}^{a-b}$

Answer 1

${\log }_{x}x-{\log }_{x}y=a⟹{\log }_x\frac{x}{y}=a⟹\frac{x}{y}=x^a$

${\log }_{x}y-{\log }_{x}z=b⟹{\log }_x\frac{y}{z}=b⟹\frac{y}{z}=x^b$

${\log }_{x}z-{\log }_{x}x=c⟹{\log }_x\frac{z}{x}=c⟹\frac{z}{x}=x^c$

$(\frac{x}{y}{)}^{b-c}\times (\frac{y}{z}{)}^{c-a}\times (\frac{z}{x}{)}^{a-b}=x^{ab-ac+bc-ab+ca-bc}=x^0=1$